Doubt regarding rotations in spinor space

I am currently doing an advanced course in Quantum Mechanics. This current doubt of mine, I am unable to clarify it properly. It follows as:

Spin 1/2 particles reside in 2dim-Hilbert space( Spinor Space)...However, we talk about rotations of states in this space where the angle of rotation is measured w.r.t Euclidean Space and we also build a Rotational Operator in this 2dim-space. My doubt is that why should spin states which reside in their own Hilbert Space respond to rotations that are carried in 3-dim Eucliean Space.? I was struck with this . Can you please help me to solve this ?

Well to begin with, be clear on one thing.. a Hilbert space is infinite dimensional. Spin space is 2-dimensional, i.e. not a Hilbert space.

To say a spin-1/2 particle resides in spin space is a bit of an exaggeration. Its wavefunction is first and foremost, like all wavefunctions, a function of x and t. A spinor wavefunction has two components, whereas the vector wavefunction for a spin-1 particle has three. An important difference between the two is that the three components of the vector can be directly associated with the x, y and z axes, whereas the two components of a spinor cannot, and in this sense "lie in a different space."

However, thanks to group theory (the two-to-one mapping of the SU(2) rotations in spin space to the SO(3) rotations in position space) a three-D rotation acts in a well-defined way on the two components of the spinor.

Wikipedia admits any number of dimensions, but then adds, "Hilbert spaces arise naturally and frequently in mathematics, physics, and engineering, typically as infinite-dimensional function spaces."

What about Hilbert himself? Courant and Hilbert introduce the term when discussing the function space of solutions of a hyperbolic PDE. I don't imagine David Hilbert would call a two-dimensional space a "Hilbert space"!

There is some ambiguity in the precise definition of "Hilbert Space", at least in some of the older literature. Some include only the infinite-dimensional case and others include finite-dimensional spaces too. What I think all definitions have in common is that a Hilbert space always has the L2 norm. For example, the classic "Principles of Mathematical Analysis" by Walter Rudin has this in its closing sentence:

... L2(μ) may be regarded as an infinite-dimensional euclidean space (the so-called "Hilbert space")...

But in my own terminology, spin spaces are Hilbert spaces because they have an L2 norm.

Sorry to jump in but a Hilbert space is simply a real or complex complete inner product space. For example ##l_2## is a Hilbert space under the usual inner product ##(x,y) = \sum \bar{x}_n y_n##. Another standard example is gotten by equipping ##\mathbb{R}^{n}## with a Borel measure ##\mu## and taking the set of all square-integrable complex-valued ##\mu##-measurable functions on ##\mathbb{R}^{n}## with the inner product ##(f,g) = \int \bar{f}(x)g(x)d\mu##. This is of course just ##L^2(\mathbb{R}^{n},d\mu)##.

There is no restriction on the dimensionality of the space in full generality but that's just terminology :)